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Coarse and Fine-Grained Aspects of Gravitational Entropy

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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Astrophysics, Cosmology, and Black Holes".

Deadline for manuscript submissions: closed (16 January 2026) | Viewed by 7678

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Editors

Dr. Juan F. Pedraza

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Guest Editor

Instituto de Física Teórica UAM/CSIC, Calle Nicolás Cabrera 13-15, 28049 Madrid, Spain
Interests: string theory; quantum gravity; holography; quantum information theory; black hole physics

Dr. Andrew Svesko

E-Mail Website
Guest Editor

Department of Mathematics, King’s College London, London WC2R 2LS, UK
Interests: gravitational holography, entanglement entropy, and complexity; quantum gravity; braneworld holography

Dr. Manus R. Visser

E-Mail Website
Guest Editor

Institute for Mathematics, Astrophysics and Particle Physics, Radboud Center for Natural Philosophy, Radboud University, 6525 AJ Nijmegen, The Netherlands
Interests: theoretical physics; gravity; black holes; holography

Special Issue Information

Dear Colleagues,

Gravitational entropy remains one of the most intriguing and fruitful topics in modern theoretical physics, lying at the interface of quantum mechanics, gravity, thermodynamics, information theory, and condensed matter physics. This proposed Special Issue aims to explore the coarse and fine-grained aspects of gravitational entropy, fostering dialogue across diverse subfields of research. By bringing together leading experts, we seek to present a comprehensive collection of articles that enhance our understanding of gravitational entropy in its various forms, emphasizing both foundational approaches and recent advances.

The Special Issue will cover a broad range of topics, including, but not limited to, the following:

Black hole thermodynamics and beyond;
Extremal and near-extremal black holes;
Hawking radiation and the island formula;
Entropy inequalities and covariant entropy bounds;
Entropic origins of gravity;
Bit threads and holographic entanglement entropy;
Alternative measures of entropy in holography;
Algebraic approach to entropy;
Quantum reference frames and edge modes;
Stringy models and microstate counting.

The Special Issue aims to

Foster interdisciplinary collaboration among researchers working on gravitational entropy across various theoretical frameworks;
Highlight recent advances and ongoing debates in the field, especially concerning the interplay between quantum mechanics and gravity;
Provide a comprehensive resource that can serve as a reference for students and researchers interested in the multifaceted landscape of gravitational entropy.

By addressing both coarse and fine-grained aspects of gravitational entropy, this Special Issue will contribute significantly to the discourse surrounding one of the most fundamental questions in theoretical physics: how do we understand the nature of entropy in a gravitational context? We invite submissions from researchers worldwide to enrich this dialogue and advance our collective understanding of gravitational entropy.

Dr. Juan F. Pedraza
Dr. Andrew Svesko
Dr. Manus R. Visser
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-anonymized peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

gravitational entropy
black hole thermodynamics
entanglement entropy
entropy in holography

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Published Papers (6 papers)

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Research

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15 pages, 377 KB

Open AccessFeature PaperArticle

Planar Black Holes and Entanglement Entropy in Analog Gravity Models

by Neven Bilic and Tobias Zingg

Entropy 2026, 28(3), 345; https://doi.org/10.3390/e28030345 - 19 Mar 2026

Viewed by 682

Abstract

Via constructing an explicit Lagrangian for which the perturbation equations are analogs of a scalar field propagating in a planar black-hole space–time, it is found that all planar black holes conformal to a Painlevé–Gullstrand-type line element can be realized as analog metrics. We also introduce the concept of holographic entanglement entropy for planar black-hole space–times. This is valid for an arbitrary choice of conformal and blackening factor, thereby vastly extending the number of known examples of explicitly known analog metrics. Full article

(This article belongs to the Special Issue Coarse and Fine-Grained Aspects of Gravitational Entropy)

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20 pages, 581 KB

Open AccessArticle

Population–Coherence Routes to Purity in Page-Type Models of Black-Hole Evaporation

by José J. Gil

Entropy 2026, 28(3), 263; https://doi.org/10.3390/e28030263 - 27 Feb 2026

Viewed by 637

Abstract

We revisit the black-hole information problem from the viewpoint of a population–coherence decomposition of density-matrix purity. Building on a previously developed formalism for n-dimensional density matrices, we characterize each state by a normalized global purity index and two complementary indices, which quantify the contributions of level populations and coherences. This yields a simple quadratic relation and a geometric representation in a “population–coherence plane”, where different routes to purity can be distinguished. In the two-level case, we construct explicit families of states with identical spectra and global purity but opposite internal structure, realizing population-dominated and coherence-dominated routes. We then apply this framework to a standard Page-type evaporation model without an explicit Hamiltonian, in which a black hole and its Hawking radiation form a bipartite pure state with varying Hilbert-space dimensions. Using known results for typical reduced states in large dimensions, we analyze the behavior of population and coherence components of purity along the evaporation process. Under the physically motivated requirement that, in this energy-free setting, the radiation populations remain nearly uniform in the chosen basis, we show that the late-time recovery of purity must be coherence-dominated: the global purity of the radiation approaches unity while the population index stays small and the coherence index carries essentially all the purity. Full article

(This article belongs to the Special Issue Coarse and Fine-Grained Aspects of Gravitational Entropy)

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12 pages, 264 KB

Open AccessArticle

Timelike Thin-Shell Evolution in Gravitational Collapse: Classical Dynamics and Thermodynamic Interpretation

by Axel G. Schubert

Entropy 2026, 28(1), 96; https://doi.org/10.3390/e28010096 - 13 Jan 2026

Viewed by 902

Abstract

This work explores late-time gravitational collapse using timelike thin-shell methods in classical general relativity. A junction surface separates a regular de Sitter interior from a Schwarzschild or Schwarzschild–de Sitter exterior in a post-transient regime with fixed exterior mass M (ADM for [...] Read more.

Λ_{+} = 0

), modelling a vacuum–energy core surrounded by an asymptotically classical spacetime. The configuration admits a natural thermodynamic interpretation based on a geometric area functional

S_{shell} \propto R^{2}

and Tolman redshift, both derived from classical junction conditions and used as an entropy-like coarse-grained quantity rather than a fundamental statistical entropy. Key results include (i) identification of a deceleration mechanism at the balance radius

R_{thr} = {(3 M / Λ_{-})}^{1 / 3}

for linear surface equations of state

p = w σ

; (ii) classification of the allowable radial domain

V (R) \leq 0

for outward evolution; (iii) bounded curvature invariants throughout the shell-supported spacetime region; and (iv) a mass-scaled frequency bound

f_{c} R_{S} \leq ξ / (3 \sqrt{3} π)

for persistent near-shell spectral modes. All predictions follow from standard Israel junction techniques and provide concrete observational tests. The framework offers an analytically tractable example of regular thin-shell collapse dynamics within classical general relativity, with implications for alternative compact object scenarios. Full article

(This article belongs to the Special Issue Coarse and Fine-Grained Aspects of Gravitational Entropy)

17 pages, 386 KB

Open AccessArticle

A Horizon-as-Apparatus Model That Reproduces Black Hole Thermodynamics

by Daegene Song

Entropy 2025, 27(8), 859; https://doi.org/10.3390/e27080859 - 14 Aug 2025

Viewed by 1745

Abstract

We present a measurement-driven model in which the black hole horizon functions as a classical apparatus, with Planck-scale patches acting as detectors for quantum field modes. This approach reproduces the Bekenstein–Hawking area law

S_{BH} = \frac{A}{4 ℓ_{p}^{2}}

and provides [...] Read more.

S_{BH} = \frac{A}{4 ℓ_{p}^{2}}

and provides a concrete statistical interpretation of the

1 / 4

factor, while adhering to established principles rather than deriving the entropy anew from first principles. Each patch generates a thermal ensemble (∼0.25 nat per mode), and summing over area-scaling patches yields the total entropy. Quantum simulations incorporating a realistic Hawking spectrum produce

⟨ S_{k} ⟩ = 0.257

nat (3% above 0.25 nat), and we outline testable predictions for analogue systems. Our main contribution is the horizon-as-apparatus mechanism and its information-theoretic bookkeeping. Full article

(This article belongs to the Special Issue Coarse and Fine-Grained Aspects of Gravitational Entropy)

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34 pages, 7929 KB

Open AccessReview

Interior Microstates and Black Hole Entropy

by Martin Sasieta

Entropy 2026, 28(4), 408; https://doi.org/10.3390/e28040408 - 3 Apr 2026

Viewed by 1193

Abstract

Semiclassical gravity admits a vast set of candidate black hole interior states, raising the question of which of these correspond to independent quantum microstates that account for black hole entropy. In this review, we survey several explicit constructions of black hole interior microstates in AdS₂ holography and AdS/CFT and assess whether they furnish bases of the black hole Hilbert space. We further highlight the settings in which non-perturbative effects in the gravitational path integral, captured by spacetime wormholes, resolve the resulting overcounting and reproduce the black hole entropy from state counting. Full article

(This article belongs to the Special Issue Coarse and Fine-Grained Aspects of Gravitational Entropy)

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75 pages, 1361 KB

Open AccessReview

Matrix Quantum Mechanics and Entanglement Entropy: A Review

by Jackson R. Fliss and Alexander Frenkel

Entropy 2026, 28(1), 58; https://doi.org/10.3390/e28010058 - 31 Dec 2025

Cited by 1 | Viewed by 1327

Abstract

We review aspects of entanglement entropy in the quantum mechanics of

N \times N

matrices, i.e., matrix quantum mechanics (MQM), at large N. In doing so, we review standard models of MQM and their relation to string theory, D-brane physics, and emergent [...] Read more.

We review aspects of entanglement entropy in the quantum mechanics of

N \times N

matrices, i.e., matrix quantum mechanics (MQM), at large N. In doing so, we review standard models of MQM and their relation to string theory, D-brane physics, and emergent non-commutative geometries. We overview, in generality, definitions of subsystems and entanglement entropies in theories with gauge redundancy and discuss the additional structure required for definining subsystems in MQMs possessing a

U (N)

gauge redundancy. In connecting these subsystems to non-commutative geometry, we review several works on ‘target space entanglement,’ and entanglement in non-commutative field theories, highlighting the conditions in which target space entanglement entropy displays an ‘area law’ at large N. We summarize several example calculations of entanglement entropy in non-commutative geometries and MQMs. We review recent work in connecting the area law entanglement of MQM to the Ryu–Takayanagi formula, highlighting the conditions in which

U (N)

invariance implies a minimal area formula for the entanglement entropy at large N. Finally, we make comments on open questions and research directions. Full article

(This article belongs to the Special Issue Coarse and Fine-Grained Aspects of Gravitational Entropy)

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